Slowly-rotating stars and black holes in dynamical Chern-Simons gravity
Chern-Simons (CS) modified gravity is an extension to general relativity (GR) in which the metric is coupled to a scalar field, resulting in modified Einstein field equations. In the dynamical theory, the scalar field is itself sourced by the Pontryagin density of the space-time. In this paper, the coupled system of equations for the metric and the scalar field is solved numerically for slowly-rotating neutron stars described with realistic equations of state and for slowly-rotating black holes. An analytic solution for a constant-density nonrelativistic object is also presented. It is shown that the black hole solution cannot be used to describe the exterior spacetime of a star as was previously assumed. In addition, whereas previous analysis were limited to the small-coupling regime, this paper considers arbitrarily large coupling strengths. It is found that the CS modification leads to two effects on the gravitomagnetic sector of the metric: Near the surface of a star or the horizon of a black hole, the magnitude of the gravitomagnetic potential is decreased and frame-dragging effects are reduced in comparison to GR. In the case of a star, the angular momentum , as measured by distant observers, is enhanced in CS gravity as compared to standard GR. For a large coupling strength, the near-zone frame-dragging effects become significantly screened, whereas the far-zone enhancement saturate at a maximum value . Using measurements of frame-dragging effects around the Earth by Gravity Probe B and the LAGEOS satellites, a weak but robust constraint is set to the characteristic CS lengthscale, km.
Gravity, one of the four fundamental forces of nature, is elegantly described by Einstein’s theory of general relativity (GR). Nearly a century after its discovery, GR has successfully passed the more and more subtle and precise tests that it has been submitted to (for a review, see for example Ref. Will (2006)). Nevertheless, Einstein’s theory is probably not the final word on gravity. We expect that a more fundamental theory unifying all forces should be able to describe not only gravity but also quantum phenomena that may take place, for example, at the center of black holes. Since gravity does seem to describe nature quite faithfully at the energy and length scales that are accessible to us, we expect that it may be the low-energy limit of such a fundamental theory. If this is the case, gravity should be described by an effective theory, for which the action contains higher-order curvature terms than standard GR, the effect of which can become apparent in strong gravity situations.
One such theory is modified Chern-Simons (CS) gravity (for a review on the subject, see Ref. Alexander and Yunes (2009)). In this theory, the metric is coupled to a scalar field through the Pontryagin density (to be defined below). The dynamical and nondynamical versions of the theory (which in fact are two separate classes of theories), then differ in the prescription for the scalar field. In nondynamical CS gravity, the scalar field is assumed to be externally prescribed. It is often taken to be a linear function of coordinate time (the so-called “canonical choice”), which selects a particular direction for the flow of time Jackiw and Pi (2003), and induces parity violation in the theory. Nondynamical CS gravity then depends on a single free parameter, which has been constrained with measurement of frame-dragging on bodies orbiting the Earth Smith et al. (2008), and with the double-binary-pulsar Yunes and Spergel (2009); Ali-Haimoud (2011). Nondynamical CS theory is quite contrived as a valid solution for the spacetime must satisfy the Pontryagin constraint ; it should therefore rather be taken as a toy model used to gain some insight in parity-violating gravitational theories.
Dynamical CS gravity, which is the subject of the present paper, is a more natural theory where the scalar field itself is given dynamics (even though arbitrariness remains in the choice of the potential for the scalar field). The scalar field evolution equation is sourced by the Pontryagin density, which is non-vanishing only for spacetimes which are not reflection-invariant, as is the case in the vicinity of rotating bodies. Contrary to the non-dynamical theory, though, dynamical CS gravity is not parity breaking, but simply has different solutions than GR for spacetimes which are not reflection-invariant (see discussion in Sec. 2.4 of Ref. Alexander and Yunes (2009)). Dynamical CS gravity has only recently received some attention, as it is more complex than the nondynamical version. Refs. Yunes and Pretorius (2009); Konno et al. (2009) computed the CS correction to the Kerr metric in the slow-rotation approximation. Ref. Sopuerta and Yunes (2009) studied the effect on CS gravity on the waveforms of extreme- and intermediate-mass ratio inspirals. Recently, Ref. Yunes et al. (2010), proposed a solution for the spacetime inside slowly-rotating neutron stars, assuming that the solution outside the star was identical to that of a black hole of the same mass and angular momentum. All the aforementioned studies were done in the small-coupling limit, i.e. considering the CS modification as a perturbation around standard GR.
The two main points of the present work are as follows. First, we show that, in contrast with GR, the spacetime around a slowly-rotating relativistic star is different from that of a slowly-spinning black hole in CS gravity, owing to different boundary conditions (regularity conditions at the horizon for a black hole versus continuity and smoothness conditions at the surface of a star). Second, we solve for the CS modification to the metric and the scalar field simultaneously, in the fully-coupled case (nonperturbative with respect to the CS coupling strength), for a slowly-rotating star or black hole. Our motivation in doing so is that frame-dragging effects are difficult to measure and are not highly constrained; it is therefore still possible that they differ significantly from the GR prescription. The solutions obtained would be exact (modulo the slow-rotation approximation) if CS gravity were taken as an exact theory. If one asumes the CS action is only the truncated series expansion of an exact theory, then our solutions are only meaningful at the linear order in the CS coupling strength.
As shown in previous works, the CS correction only affects the gravitomagnetic sector of the metric at leading order in the slow-rotation limit. We find that for both stars and black holes, the CS correction leads to a suppression of frame-dragging at a distance of a few stellar radii, or a few times the black hole horizon radius. This suppression is perturbative in the small-coupling regime but can become arbitrarily large in the nonlinear regime, asymptotically leading to a complete screening of frame-dragging effects near the star or black hole in the large-coupling-strength limit. Far from the boundary of a star, the magnitude of frame-dragging effects is enhanced: we find a correction to the metric component at large radii, with . This means that for a given angular rotation rate , the angular momentum measured by distant observers is enhanced as compared to standard GR. We evaluate the correction to the angular momentum as a function of the coupling strength and find that it increases quadratically with the coupling parameter in the small-coupling regime, and asymptotes to a constant value in the large-coupling regime.
These results are obtained using an analytic approximation for constant-density nonrelativistic objects, and confirmed with a numerical solution for neutron stars described with realistic equations of state. The black hole solution is computed numerically and checked against known analytic solutions in the small coupling regime.
Finally, using measurements of frame-dragging effects around the Earth, we set a weak but robust constraint to the CS characteristic lengthscale, km. We argue that this bound is the only current astrophysical constraint to the theory.
This paper is organized as follows: in Sec. II, we review the theory of dynamical Chern-Simons gravity and define our notation. In Sec. III, we lay out the general formalism to compute the CS scalar field and the metric in CS gravity, for a slowly rotating object. We provide analytic expressions for the exterior solution in some limiting cases in Sec. IV, as a well as an analytic solution for the full spacetime for nonrelativistic constant-density stars. We present the results of our numerical computations for realistic neutron stars in Sec. V and for black holes in Sec. VI. We discuss constraints to the theory in Sec. VII and conclude in Sec. VIII.
Throughout this paper we use geometric units . We adopt the conventions of Ref. Misner et al. (1973) for the signature of the metric, Riemann and Einstein tensors.
Ii Dynamical Chern-Simons gravity
For a review on Chern-Simons modified gravity, we refer the reader to Ref. Alexander and Yunes (2009). Here we simply recall the main equations and results and define our notation.
We consider the following action defining the modified theory 111The conversion from the notation used in Ref. Alexander and Yunes (2009) (AY09) to that of the present work is given by , , . The conversion from the notation of Ref. Smith et al. (2008) (who define the Riemann tensor with an opposite sign) is given by .:
In Eq. (1), the first term contains the standard Einstein-Hilbert action and the matter contribution with lagrangian density . The second term is the CS modification, which only depends on the dimensionfull coupling constant (which has dimensions of length; its relation to the parameter of Ref. Yunes and Pretorius (2009) is ), and of course on the shape of the potential for the dimensionless CS scalar field . The Pontryagin density is given by the contraction of the Riemann tensor with its dual:
where is the four-dimensional Levi-Civita tensor (with convention in a right-handed orthonormal basis). The equations of motion resulting from the modified action are the following:
(i) The modified Einstein field equations:
where is the Einstein tensor, is the matter stress-energy tensor,
is a four-dimensional generalization of the Cotton-York tensor, and
is the stress-energy tensor associated with the scalar field .
(ii) The evolution equation for the scalar field :
where is the usual covariant d’Alembertian operator.
In general, the fundamental theory from which CS gravity arises should predict a shape for the potential . This would introduce an additional free parameter (or several parameters) in addition to . Since our goal here is mainly to study the effect of the coupling strength , we follow previous works and assume for simplicity.
Iii Slowly-rotating relativistic stars in Chern-Simons gravity
We consider a stationary, axially symmetric system. This means that there exist a time coordinate and an angular coordinate such that the metric components do not depend on nor ; if and are the two remaining spatial coordinates, we have
We moreover assume that the line element satisfies an additional discrete symmetry: we suppose that the effect of inversion of the azimuthal angle is identical to that of inverting time, , i.e.
Note that one could clearly not have made this assumption in nondynamical CS gravity where the externally prescribed scalar field defines the flow of time (for the “canonical” choice of scalar field). The validity of this assumption in dynamical CS gravity will be justified a posteriori by checking that solutions do exist with this discrete symmetry (however, it does not guarantee the uniqueness of the solutions found). With this assumption, we find that222In GR, one does not need the additional assumption (9) to obtain (10), see proof in Section 7.1 of Ref. Wald (1984). The proof of Ref. Wald (1984) makes use of the GR Einstein field equation, however, and can therefore not be carried over to CS gravity. One could also retain the a priori non-zero and metric components and show that the CS field equations lead them to vanish. We have not chosen this path in order ot avoid excessive algebra.
By an appropriate choice of coordinate transformations of the form for , one may rewrite the line element in the form Hartle and Sharp (1967)
where we have removed the tildes for clarity and denoted , .
iii.2 Stress-energy tensor
We consider an ideal fluid in solid rotation with angular rate , i.e. such that the 4-velocity is of the form
To evaluate , we use the normalization condition , i.e.
The stress-energy tensor of a perfect fluid is then obtained from
where and are the density and pressure of the fluid as measured in its local rest frame.
iii.3 Slow-rotation expansion
We consider a star333In this paper we use the word ‘star’ to refer to a general (non-empty) astrophysical object. of radius in solid rotation with angular velocity . The star is in slow rotation if . In what follows we shall expand the metric functions and as well as the density and pressure fields in powers of .
Under the change of coordinates , the angular velocity changes sign, , and therefore remain unchanged, while and therefore change sign. Similarly, considerations of the transformation properties of the stress-energy tensor show that and remain unchanged. Since the field equation expressed in the coordinate system are the same as those in the coordinate system for a fluid with angular velocity (because there is no functional dependence on ), we conclude that and are even functions of whereas is an odd function of .
In the slow-rotation approximation, we only keep the lowest-order contribution to each function. From the previous discussion, this means that we keep only the terms in and and the term in . The next-order contributions are of relative order in CS gravity, where is an unknown function of the coupling strength. It is beyond the scope of this paper to evaluate , but we emphasize that our slow-rotation approximation is only valid as long as .
To lowest order, the velocity of the fluid becomes
Moreover, the Pontryagin density is of order (we give the exact expression below). Therefore, the scalar field is also of order [from Eq.(6) with ]. As a consequence, the components of the scalar field stress-energy tensor [Eq. (5)] are of order and need not be considered at this level of approximation, i.e. the energy of the scalar field does not curve space-time to first order in the rotation rate.
iii.4 Stellar structure equations for and
The leading-order contribution to the metric components which are even in can be obtained in the non-rotating case, . In that case the system is spherically symmetric and the Pontryagin density vanishes. The scalar field being unsourced, the Einstein field equations are identical to those of standard GR. By spherical symmetry, and are functions of only. By rescaling the radial coordinate, one can moreover set to unity. The functions and then satisfy the usual relativistic stellar structure equations Misner et al. (1973):
being the mass enclosed within a radius . Inside the star, the function is the solution of the equation
where the pressure must satisfy the equation of hydrostatic equilibrium
Outside the star, we have , and
where is the total mass of the star.
iii.5 Gravitomagnetic sector
iii.5.1 General equations
Let us now consider the component of the metric. Evaluating the component of the modified field equation to lowest order in , we find, in agreement with Ref. Yunes et al. (2010) and Refs. Yunes and Pretorius (2009); Konno et al. (2009) outside the star:
In addition, the evolution equation for the CS scalar field becomes, in the time-independent case and to first order in :
where, to lowest order, the Pontryagin density is given by
Finally, we also note that all other non-diagonal components of the Einstein, “C” and stress-energy tensors vanish. Therefore the metric (11) is indeed a valid solution of the modified Einstein field equations provided we solve for , and as described above.
iii.5.2 Multipole expansion
Following Ref. Konno et al. (2009), we decompose on the basis of Legendre polynomials and on the basis of their derivatives:
Eqs. (21) and (22) can then be rewritten as an infinite set of coupled differential equations for the coefficients and . Each pair of equations is independent in the sense that the -th multipole of only couples to the -th multipole of Konno et al. (2009). The only sourced multipole is , and therefore and for (this follows from the system being second order in and , and the requirement that and vanish at large radii and be bounded at the stellar center). The functions and are the solutions of the following system
One can decompose into two pieces: a part that would be present in standard GR, , that can be obtained by setting in Eq. (26), and a correction (not necessarily small) , which is the solution of Eq. (26) with . Note that it is the full that sources the CS scalar field in Eq. (27).
Before proceeding further, we shall discuss boundary conditions at the surface of the star. First, the function must be continuous at the boundary of the star. In addition, integrating Eq. (26) between and gives us a jump condition for the derivative of at the stellar surface:
For neutron stars, the surface density is nearly vanishing [in the sense that ], and is (nearly) continuous at the surface. For constant density objects, however, there is a jump in the derivative of at the surface. Finally, inspection of Eq. (27) shows that all coefficients are bounded (although potentially discontinuous), and therefore both and must be continuous at the stellar surface, in all cases.
iii.5.3 Simplification outside the star
Outside the star, and Eq. (26) can be integrated once and simplified to
where is a constant of integration. is also the total angular momentum of the star as measured by observers in the asymptotically flat far zone, as we shall discuss in Sec. III.6. We write , where is the value of in standard GR and is the correction (not necessarily perturbative) that arises in CS gravity.
where we have defined the dimensionless coupling strength , to be used repeatedly in the remainder of this paper:
Physically, is of the order of the ratio of the CS lengthscale to the dynamical timescale of the system. It is important to notice that the definition of depends on the system considered through its average density.
iii.6 A note on the angular momentum and moment of inertia
There are several possible definitions for angular momentum in GR, and we therefore specify the definition that we use here.
In spherical polar coordinates, and using the notation of Eq. (11), this corresponds to
We therefore see that the constant of integration in Eq. (29) corresponds to the total angular momentum of the star or black hole as measured by distant observers.
The moment of inertia is an ill-defined quantity for relativistic systems, as in general the angular momentum of a body in solid rotation may not scale linearly with the angular velocity. However, in the slow-rotation approximation, the angular momentum does scale linearly with , to first order. We can therefore define the relativistic generalization of the Newtonian moment of inertia by Hartle (1967)
In general the moment of inertia will depend on the mass and equation of state of the considered object. In Chern-Simons gravity, it will also depend on the coupling constant .
In GR, there exists a simple integral formula for the moment of inertia Hartle (1967):
This formula is a priori valid only in GR, and does not necessarily hold in modified gravity theories. Equation (35) is indeed a local definition, whereas we have defined angular momentum from its imprint on the spacetime in the far-field.
Multiplying Eq. (36) by , integrating form 0 to , and using the jump condition for at the surface of the star, Eq. (28), in conjunction with Eq. (29), we recover Eq. (35). We emphasize that this integral formula is not a definition of the moment of inertia and would not necessarily be valid in other modified gravity theories.
Iv Analytic approximate solutions
Before tackling the full numerical solution of the problem, we give a few analytic results in some simple cases.
iv.1 Analytic exterior solution in the small coupling limit
where and are constants of integration. The requirement that remains finite at infinity implies that . In Refs. Yunes and Pretorius (2009) and Konno et al. (2009) where a rotating black hole was studied, was also (rightly) set to zero so (as well as ) remains finite at the horizon. In our case, however, since outside the star, the homogeneous solution proportional to is well behaved everywhere outside the star, and a priori (in fact, we shall show that for nonrelativistic objects). The integration constant must be determined from the continuity and smoothness requirements for at the stellar boundary. This shows that the black hole solution cannot be used as the solution outside a star as was assumed in Ref. Yunes et al. (2010).
From Eq. (29), we then obtain outside the star (setting the additional integration constant to zero so that ):
It will be useful in what follows to write the asymptotic behavior of and at large radii up to corrections of relative order (a fortiori, these results are also valid everywhere outside the source in the nonrelativistic limit ):
Before going further, let us assess the differences of our results with those of Ref. Yunes et al. (2010), who also worked in the small coupling regime. First, as we pointed out previously, Ref. Yunes et al. (2010) set whereas we shall show below that . Therefore our solution for the scalar field is of order smaller than that of Ref. Yunes et al. (2010) at large distances. Second, and more importantly, in Ref. Yunes et al. (2010), the parameter used in Eq. (38) was set to . In reality, is determined by imposing the continuity and jump conditions for at the surface of the star. In the small coupling regime,
and is in general nonzero. We therefore have at large distances, instead of in Ref. Yunes et al. (2010). Physically, this means that the CS correction translates into a change of angular momentum as measured by distant observers, whereas the previous solution did not, strictly speaking, lead to any additional angular momentum.
iv.2 Analytic exterior solution in the nonrelativistic limit
If then the solution of Eq. (30) is
where and are integration constants. Requiring to be finite at large radii implies , whereas needs to be fixed using continuity conditions at the stellar boundary.
From Eq. (29), we then obtain outside the star (choosing the additional integration constant such that ):
If we Taylor-expand these solutions for , we should recover the analytic solution obtained in Sec. IV.1 in the small-coupling limit, in the far-field limit, (we have just reversed the order in which the limits are taken). We obtain, up to corrections of relative order :
iv.3 Analytic solution in the nonrelativistic limit for a constant density object
We have already obtained the general solution outside a nonrelativistic star in the previous section. We now need a solution inside the star to obtain the integration constants and .
Let us first consider Eq. (27) for . For a constant density object, the right-hand-side vanishes, and in the non-relativistic limit, we find
Requiring to be finite at the center of the star, we set . We therefore have . The continuity and smoothness of at the boundary therefore imply that this relation is also satisfied at . Imposing this condition with given in Eq. (42) (where we recall that ), we obtain, up to corrections of order ,
From the value of and Eq. (44) we infer the asymptotic behavior of at large radii ():
We show the radial dependence of the scalar field for a nonrelativistic constant density object in Fig. 1. We see that in the linear regime (), the scalar field increases uniformly with — for , . Increasing further eventually leads to a damping of the scalar field near the surface of the star. In the far zone, we see from Eq. (51) that the asymptotic value of plateaus to a constant value.
Let us now consider Eq. (26) for . Again, for a constant density object, the right-hand-side vanishes. Using , the equation satisfied by becomes, to first order in :
which has the general solution, up to corrections of order :
In order for to be finite at the origin, we impose . Requiring to be continuous at the boundary and its derivative to satisfy the jump condition Eq. (28), we find and
If we write (with the well-known result in the non-relativistic limit), we therefore obtain, to lowest order in ,
In the small coupling limit (), we find
Interestingly, for large values of the coupling constant, the Chern-Simons correction saturates
We show the function for a constant-density, nonrelativistic object in Fig. 2. The effect of CS gravity is to (potentially strongly) decrease the gravitomagnetic potential near the surface, and enhance its asymptotic value in the far field by a relative amount .
In the next section we will see that the qualitative features of the analytic solution are recovered in the full numerical solution.
V Numerical solution for realistic neutron stars
In the previous section we have given analytic solutions for the coupled CS scalar field-metric system in the nonrelativistic limit, for constant density objects. In the present section we provide full numerical solutions for neutron stars described by realistic equations of state.
v.1 Equations of state
Matter at nuclear densities has complex properties, and the EOS of neutron stars is not very well known. There exist two EOSs that are widely used in astrophysical simulations (see for example Ref. O’Connor and Ott (2010) for a discussion): the Lattimer-Swesty (LS) EOS Lattimer and Swesty (1991) and the Shen et al EOS Shen et al. (1998, 1998). We will use the LS EOS with nuclear incompressibility MeV (hereafter, LS220) and the Shen et al EOS (hereafter Shen). We use the EOS routines of O’Connor and Ott444Available at http://stellarcollapse.org/microphysics; we thank Evan O’Connor for providing tabulated solutions to the relativistic equations of stellar structure. O’Connor and Ott (2010) to solve the relativistic stellar structure equations. We show the resulting mass-radius relations in Fig. 3.
v.2 Numerical solution of the coupled CS equations
v.2.1 Boundary conditions
Inspection of the system given by Eqs. (21) and (27) near the origin shows that there is one well behaved solution for with and a divergent (hence unphysical) solution with . Similarly, there is one well-behaved solution for with and a divergent, unphysical solution with . The physically allowed boundary conditions at the origin are therefore
where and are integration constants to be determined.
The mathematically allowed asymptotic behaviors at infinity are similar, but the physically relevant solutions are reversed, i.e. we have
where and are integration constants, and can physically be interpreted as the total perceived angular momentum of the system.
In addition to these boundary conditions at and , the functions and must all be continuous at the boundary of the star (in principle there is a jump in the derivative of , see Eq. (28), but for realistic neutron stars the surface density is 7 orders of magnitude than the mean density, so is continuous up to small corrections).
v.2.2 Shooting method
We integrate the coupled system given by Eqs. (21) and (27) with a second-order implicit Euler method from to , and from down to (specifically, we consider the functions and from to ; these functions vanish at and their derivatives at are proportional to the constants and ). The linearity of the system allows to find the appropriate constants of integration at and with a shooting method by requiring continuity and smoothness at .
In this section we illustrate our numerical results in several figures, and compare them to our analytic solution of Sec. IV.3.
v.3.1 Scalar field
First, in Fig. 5, we show the scalar field as a function of radius, for several values of the dimensionless CS coupling parameter . A qualitative difference of this work with that of Ref. Yunes et al. (2010) is that we have properly enforced the continuity and the smoothness of the CS scalar field at the stellar boundary (see Fig. 4 of Ref. Yunes et al. (2010) for a comparison). In the linear regime , increases uniformly with . For , more complex nonlinear behaviors appear (the system solved being equivalent to a fourth-order ODE, such behaviors can be expected), but the overall effect is to suppress near the stellar surface. The far-field behavior of plateaus to an asymptotic limit, as we found in our analytic approximation.
v.3.2 Gravitomagnetic sector and moment of inertia
In Fig. 6, we show the gravitomagnetic sector of the metric through the function . An essential difference of our solution with previous work is that we have at large radii, instead of .
In Fig. 7 we show the relative change in moment of inertia induced by the CS modification, , as a function of the dimensionless coupling strength . We recall that the moment of inertia is defined as a , where . We also plot the result of our analytic approximation for a nonrelativistic constant density star. Although the overall normalization is off by nearly an order of magnitude, we see that the trends predicted by our simple approximation are indeed recovered in the full numerical result. In the limit of small CS coupling, we have . For large values of the dimensionless coupling parameter, the CS correction to the moment of inertia asymptotes to a constant value.
We show the dependence of on neutron star compactness in the small coupling limit in Fig. 8. Again, our very simple analytic result is underestimating the exact numerical result, but the trend is roughly respected for realistic neutron star masses.
Finally, in Fig. 9 we show the extrapolated asymptotic value of in the limit of large coupling parameter, as a function of neutron star compactness. It depends approximately linearly on neutron star compactness, as we found in our analytic solution. To obtain this value we have computed for several values of the coupling strength up to and fitted — we could not go much beyond as the homogeneous solutions contain terms of order that quickly become very large and lead to large numerical errors, see equations of Sec. IV.2.
Our numerical results are therefore in very good qualitative agreement with our analytic approximation for constant-density nonrelativistic objects. Given that a neutron star density profile is far from constant and that neutron stars are compact, it is not surprising that the quantitative agreement is not perfect. We can understand why the analytic approximation systematically underestimates the correct result. A realistic neutron star is much more centrally concentrated than a constant density object. The “effective” radius of the star (defined, for example, as the radius containing 75% of the mass), is always significantly smaller than the actual stellar radius. A neutron star is therefore more compact in its central regions than if it had a constant density. For example, the mean density of a 1 neutron star inside the sphere containing 75% of its mass is about 70% larger than its overall mean density.
Vi Slowly-rotating black hole in Chern-Simons gravity
In this section we compute the solution for the scalar field and the metric of a slowly-rotating black hole in CS gravity, for an arbitrary coupling constant.
The equation satisfied by the CS scalar field for a slowly-rotating black hole is the same as that outside a slowly-rotating star, Eq. (30). Using , and defining the dimensionless coupling parameter in analogy with Eq. (31) with the substitution
the equation for the scalar field becomes
where dots denote differentiation with respect to .
There are two qualitative differences between the black hole case and the stellar case.
Firstly, in the case of a slowly-rotating black hole, the angular momentum is just a given parameter (as well as the black hole mass), independently of the gravity theory chosen. Of course, is the angular momentum of the object that collapsed into a black hole, in which case its value does depend on the gravity theory chosen; once the black hole is formed, however, th